Question

Two law firms in a community handle all the cases dealing with consumer suits against companies in the area. The Abercrombie firm takes 40% of all suits, and the Olson firm handles the other 60%. The Abercrombie firm wins 70% of its cases, and the Olson firm wins 60% of its cases.
a. Develop a probability tree showing all marginal, conditional, and joint probabilities.
b. Develop a joint probability table.
c. Using Bayes’ rule, determine the probability that the Olson firm handled a particular case, given that the case was won.

Answer 1

Answer:

Part A: Diagram

Psrt B:

Joint Probability Table

Firms                                     Success                           Failure        

Abercrombie                          0.28                                   0.12

Oslon                                      0.36                                  0.24

Part C :  P (O/S)       =0.5625

Explanation:

The probability tree can be drawn as follows

Part A:

║⇒⇒P (A) = 0.4⇒⇒⇒⇒║⇒⇒⇒⇒P (S/A)= 0.7⇒⇒⇒⇒ P (A∩S)= 0.28

║                                     ║

║                                     ║⇒⇒⇒⇒ P (F/A)= 0.3⇒⇒⇒ P (A∩F)= 0.12

║

║⇒⇒⇒P (O)= 0.6⇒⇒⇒⇒║⇒⇒⇒⇒P (S/O)= 0.6⇒⇒ P (O∩S)= 0.36

                                          ║

                                          ║⇒⇒⇒P (F/O)= 0.4⇒⇒ P (O∩F)= 0.24

The marginal Probability of the two firms

P (A)= 0.4

P (O)= 0.6

Where   P (A) is the probability of Abercrombie firm

P (O)  is the probability of Olson firm

The conditional probabilities are given by

P (S/A)= 0.7

P (F/A)= 0.3

Where   P (S/A) is the conditional  probability of Success of  Abercrombie firm

P (F/A)  is the conditional  probability of failure of  Abercrombie firm

Similarly

P (S/O)= 0.6

P (F/O)= 0.4

P (S/O) is the conditional  probability of Success of  Oslon firm

P (F/O)  is the conditional  probability of failure of  Oslon firm

The probability table is given by

Firms                         Marginal          Conditional                Joint    

Abercrombie          0.4                     0.7                            0.28

                                                          0.3                           0.12

Oslon                     0.6                     0.6                            0.36

                                                        0.4                             0.24

Joint Probability Table

Firms                                     Success                           Failure        

Abercrombie                          0.28                                   0.12

Oslon                                      0.36                                  0.24

Part C :

Using Bayes Rule:

P (O/S) =   P ( O) P( S/O)/ P ( O) P( S/O)+ P (A) P(S/ A)

              = 0.6*0.6/ 0.6*0.6+0.4*0.7

              =0.36/ 0.36+0.28

              =0.5625